Pick Interpolation and Hilbert Function Spaces

Jim Agler (Author), John E. McCarthy (Author)

9781470468552

Paperback / softback, published 1 January 2002

308 pages
25.7 x 18.3 x 2.3 cm, 0.608 kg

Written in a clear, straightforward style, at a level to make it accessible to someone—a mid-level graduate student, say—who wishes to study the material in detail for the first time ... contains exercises ... as well as ... open questions. It brings the reader up to the current 'state of the art' and so will be a valuable resource for the specialist ... would be an excellent basis for a graduate seminar or topics course."" — Mathematical Reviews

""Material is wonderfully presented, and the book serves as a lovely introduction to the subject. It is written by two authorities in the field, and helps grad students get entry into an exciting, modern, and very active research area."" — Palle Jorgensen

The book first rigorously develops the theory of reproducing kernel Hilbert spaces. The authors then discuss the Pick problem of finding the function of smallest $H^\infty$ norm that has specified values at a finite number of points in the disk. Their viewpoint is to consider $H^\infty$ as the multiplier algebra of the Hardy space and to use Hilbert space techniques to solve the problem. This approach generalizes to a wide collection of spaces.

The authors then consider the interpolation problem in the space of bounded analytic functions on the bidisk and give a complete description of the solution. They then consider very general interpolation problems. The book includes developments of all the theory that is needed, including operator model theory, the Arveson extension theorem, and the hereditary functional calculus.

• Prerequisites and notation
• Introduction
• Kernels and function spaces
• Hardy spaces
• $P^2(\mu)$
• Pick redux
• Qualitative properties of the solution of the Pick problem in $H^\infty(\mathbb{D})$
• Characterizing kernels with the complete Pick property
• The universal Pick kernel
• Interpolating sequences
• Model theory I: Isometries
• The bidisk
• The extremal three point problem on $\mathbb{D}^2$
• Collections of kernels
• Model theory II: Function spaces
• Localization
• Schur products
• Parrott's lemma
• Riesz interpolation
• The spectral theorem for normal $m$-tuples
• Bibliography
• Index